Borel summability of Navier-Stokes equation in $\mathbb{R}^3$ and small time existence

Abstract

We consider the Navier-Stokes initial value problem, $$v_t - \nabla v = -\mathcal{P} [ v \cdot \nabla v \right ] + f, v(x, 0) = v_0 (x), x \in \mathbb{R}^3 $$ where $\mathcal{P}$ is the Hodge-Projection to divergence free vector fields in the assumption that $ | f |_{\mu, \beta} < \infty $ and $| v_0 |_{\mu+2, \beta} < \infty$ for $\beta \ge 0, \mu > 3$, where $$ | {\hat f} (k) | = \sup_{k \in \mathbb{R}^3} e^{\beta |k|} (1+|k|)^\mu | {\hat f} (k) |$$ and ${\hat{f}} (k) = \mathcal{F} [f (\cdot)] (k) $ is the Fourier transform in $x$. By Borel summation methods we show that there exists a classical solution in the form $$ v(x, t) = v_0 + \int_0^\infty e^{-p/t} U(x, p) dp $$ $t\in\CC$, $ \Re \frac{1}{t} > \alpha$, and we estimate $\alpha$ in terms of $| {\hat v}_0 |_{\mu+2, \beta}$ and $ | {\hat f} |_{\mu, \beta}$. We show that $| {\hat v} (\cdot; t) |_{\mu+2, \beta} < \infty $. Existence and $t$-analyticity results are analogous to Sobolev spaces ones. An important feature of the present approach is that continuation of $v$ beyond $t=\alpha^{-1}$ becomes a growth rate question of $U(\cdot, p)$ as $ p \to \infty$, $U$ being is a known function. For now, our estimate is likely suboptimal. A second result is that we show Borel summability of $v$ for $v_0$ and $f$ analytic. In particular, we obtain Gevrey-1 asymptotics results: $ v \sim v_0 + \sum_{m=1}^\infty v_m t^m $, where $ |v_m | \le m! A_0 B_0^m$, with $A_0$ and $B_0$ are given in terms of to $v_0$ and $f$ and for small $t$, with $m(t)=\lfloor B_0^{-1}t^{-1}\rfloor$, $$ | v(x, t) - v_0 (x) - \sum_{m=1}^{m(t)} v_m (x) t^m | \le A_0 m(t)^{1/2} e^{-m(t)} $$